hypergeometric equation

SearchAdvancedHelp

(0.004 seconds)

41—50 of 124 matching pages

41: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

… ►

17.8.8 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{b^2,b^2/c}{q,cq};q^2,c{q}^2/b^2)=\frac{1}{2}\frac{{(q^2,q{b}^2,q/b^2,cq/b^2;q^2)}_{\mathrm{\infty }}}{{(cq,c{q}^2/b^2,q^2/b^2,c/b^2;q^2)}_{\mathrm{\infty }}}\left(\frac{{(c\sqrt{q}/b;q)}_{\mathrm{\infty }}}{{(b\sqrt{q};q)}_{\mathrm{\infty }}}+\frac{{(-c\sqrt{q}/b;q)}_{\mathrm{\infty }}}{{(-b\sqrt{q};q)}_{\mathrm{\infty }}}\right),$$ .

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Source:

Verma and Jain (1983, (3.9))

Referenced by:

§17.6(i), §17.8, Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/17.8.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

Suggested 2019-10-19 by Slobodan Damjanovic

See also:

Annotations for §17.8, §17.8 and Ch.17

…

42: 13.27 Mathematical Applications

§13.27 Mathematical Applications

►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. … ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).

43: 15.6 Integral Representations

… ►

15.6.1 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-b\right)}{\int }_0^1\frac{t^{b-1}{(1-t)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }b>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)

Permalink:

http://dlmf.nist.gov/15.6.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

… ►

15.6.6 $$𝐅(a,b;c;z)=\frac{1}{2\pi \mathrm{i}\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(b+t\right)\mathrm{\Gamma }\left(-t\right)}{\mathrm{\Gamma }\left(c+t\right)}{(-z)}^{t}dt,$$ ; $a,b\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

A&S Ref:

15.3.2 (modified)

Referenced by:

§15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)

Permalink:

http://dlmf.nist.gov/15.6.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

►

15.6.7 $$𝐅(a,b;c;z)=\frac{1}{2\pi \mathrm{i}\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(b+t\right)\mathrm{\Gamma }\left(c-a-b-t\right)\mathrm{\Gamma }\left(-t\right){(1-z)}^{t}dt,$$ ; $a,b,c-a,c-b\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)

Permalink:

http://dlmf.nist.gov/15.6.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

►

15.6.8 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(c-d\right)}{\int }_0^1𝐅(a,b;d;zt)t^{d-1}{(1-t)}^{c-d-1}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }d>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Proof sketch:

Follows from (15.6.9) with $\lambda =a+b$.

Referenced by:

(15.6.9), §15.6, §15.6, §15.6, §18.17(iv), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)

Permalink:

http://dlmf.nist.gov/15.6.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

►

15.6.9 $$𝐅(a,b;c;z)={\int }_0^1\frac{t^{d-1}{(1-t)}^{c-d-1}}{{(1-zt)}^{a+b-\lambda }}𝐅(\genfrac{}{}{0pt}{}{\lambda -a,\lambda -b}{d};zt)𝐅(\genfrac{}{}{0pt}{}{a+b-\lambda ,\lambda -d}{c-d};\frac{(1-t)z}{1-zt})dt,$$ ; $\lambda \in ℂ$, $\mathrm{\Re }c>\mathrm{\Re }d>0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Notes:

With $\lambda =a+b$, this equation specializes to (15.6.8).

Referenced by:

(15.6.8), §15.6, §15.6, Erratum (V1.0.18) for Equation (15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)

Permalink:

http://dlmf.nist.gov/15.6.E9

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

Addition (effective with 1.0.18):

The constraint $\lambda \in ℂ$ was added.

See also:

Annotations for §15.6 and Ch.15

…

44: Mathematical Introduction

… ►This is because $𝐅$ is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as $𝐅$ is an entire function of each of its parameters $a$, $b$, and $c$:​ this results in fewer restrictions and simpler equations. …

45: 18.26 Wilson Class: Continued

… ►

§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

►For the definition of generalized hypergeometric functions see §16.2. … ►

18.26.2 $$S_n(y^2;a,b,c)={\left(a+b\right)}_n{\left(a+c\right)}_n{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,a+iy,a-iy}{a+b,a+c};1).$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $y$: real variable and $n$: nonnegative integer

Referenced by:

§18.26(i), §18.26(ii), §18.26(iv)

Permalink:

http://dlmf.nist.gov/18.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(i), §18.26 and Ch.18

… ►

§18.26(iv) Generating Functions

►For the hypergeometric function ${}_2{}^{}F_1^{}$ see §§15.1 and 15.2(i). …

46: 15.9 Relations to Other Functions

… ►

15.9.24 $$K\left(k\right)=\frac{\pi }{2}F(\genfrac{}{}{0pt}{}{\frac{1}{2},\frac{1}{2}}{1};k^2),$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind and $k$: integer

Referenced by:

§15.9(v)

Permalink:

http://dlmf.nist.gov/15.9.E24

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.9(v), §15.9 and Ch.15

►

15.9.25 $$E\left(k\right)=\frac{\pi }{2}F(\genfrac{}{}{0pt}{}{-\frac{1}{2},\frac{1}{2}}{1};k^2),$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind and $k$: integer

Referenced by:

§15.9(v)

Permalink:

http://dlmf.nist.gov/15.9.E25

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.9(v), §15.9 and Ch.15

►

15.9.26 $$D\left(k\right)=\frac{\pi }{4}F(\genfrac{}{}{0pt}{}{\frac{1}{2},\frac{3}{2}}{2};k^2).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $D\left(k\right)$: complete elliptic integral of Legendre’s type and $k$: integer

Referenced by:

§15.9(v)

Permalink:

http://dlmf.nist.gov/15.9.E26

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.9(v), §15.9 and Ch.15

47: 20.11 Generalizations and Analogs

… ►The first of equations (20.9.2) can also be written …Similar identities can be constructed for ${}_2{}^{}F_1^{}(\frac{1}{3},\frac{2}{3};1;k^2)$, ${}_2{}^{}F_1^{}(\frac{1}{4},\frac{3}{4};1;k^2)$, and ${}_2{}^{}F_1^{}(\frac{1}{6},\frac{5}{6};1;k^2)$. …For applications to rapidly convergent expansions for $\pi$ see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). … ►The importance of these combined theta functions is that sets of twelve equations for the theta functions often can be replaced by corresponding sets of three equations of the combined theta functions, plus permutation symmetry. Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

48: Bibliography

… ►

On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

ⓘ

External Links:

ISSN 0024-1318, MathReview (James M. Horner)

Referenced by:

§13.9(i).

Encodings:

BibTeX

…

49: 13.3 Recurrence Relations and Derivatives

… ►

13.3.14 $$(a+1)zU(a+2,b+2,z)+(z-b)U(a+1,b+1,z)-U(a,b,z)=0.$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.3(i)

Permalink:

http://dlmf.nist.gov/13.3.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

…

50: 15.4 Special Cases

… ►

15.4.34 $$F(3a,a;2a;{\mathrm{e}}^{\mathrm{i}\pi /3})=\sqrt{\pi }{\mathrm{e}}^{\mathrm{i}\pi a/2}\frac{2^{2a}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{3^{(3a+1)/2}}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{3}+a\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}+a\right)\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $a$: real or complex parameter

Proof sketch:

From §15.5(ii) one obtains that $w(a)={\mathrm{e}}^{-\mathrm{i}\pi a/2}{2}^{-2a}{3}^{(3a+1)/2}F(3a,a;2a;{\mathrm{e}}^{\mathrm{i}\pi /3})\mathrm{\Gamma }\left(\frac{1}{3}+a\right)/\mathrm{\Gamma }\left(\frac{1}{2}+a\right)$ is a solution of the difference equation $(3a+5)w(a+2)-6(a+1)w(a+1)+(3a+1)w(a)=0$. Two linearly independent solutions of this difference equation are $w_1(a)=1$ and $w_2(a)=\mathrm{\Gamma }\left(\frac{1}{3}+a\right)/\mathrm{\Gamma }\left(\frac{2}{3}+a\right)$. Combining (15.6.1) with Laplace’s method (see §2.4(iii)) we obtain that $w(a)\sim \sqrt{\pi }\left(\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{a^{-1/3}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right)$, as $a\to \mathrm{\infty }$. Hence, $w(a)=\sqrt{\pi }\left(\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{w_2(a)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right)$.

Notes:

See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer.

Referenced by:

§15.4(iii), §15.4(iii), Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/15.4.E34

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.4(iii), §15.4 and Ch.15

…